Abstract:
Explicit recurrent formulas are given for the unique and stable classical solution of the characteristic mixed problem for the inhomogeneous simplest vibration equation of a bounded string. For any moment of time in the characteristic boundary conditions at the ends of the string, the oblique derivatives with time-dependent coefficients are directed along the critical characteristics of the equation. A correctness criterion of this mixed problem is derived, i.e. necessary and sufficient smoothness requirements and matching conditions the characteristic boundary conditions with the initial conditions and the string vibration equation for the existence, uniqueness and stability of its classical solutions. The derivation of matching conditions essentially uses the new concept of criterion values for the sum of the highest derivatives of the right-hand side of the equation. These results were obtained by the well-known method of auxiliary mixed problems for a semi-bounded string, `which does not require explicit periodic continuations of the mixed problems data outside their definition sets.
